Please answer question 13.2 within an hour or asap please, I provided other photos to provide context. Problem 13.2 is at the very bottom

13.3 Quantum Entanglement and the Impact of Measurement We now consider something more complicated, based on the principle of superposition. To illustrate the physics, we will make the observables something we understand: linear momentum and spin, Consider a simple experiment. At some point in time, a system with zero angular momentum decays into two spin '/2 particles moving in opposite directions Since the total angular momentum must remain zero, the only option for the spin part of the state vector is that it correspond to the singlet with 5: m5 = 0 and Isms) = '00) = % (HA \"2);! ILZ)1 H32) for the two electrons where the arrows represent the projection of l3.3 QUANTUM ENTANGLEMENTAND THE IMPACT OF MEASUREMENT Alice Bob Fig. 13.1 A source emits two spin V2 particleswith total angular momentum ofzero, traveling in opposite direction to two different detectors operated by Alice and Bob. the spinJ/2 on the zeaxis. However, the state is also described by the linear momentum for each electron. Again, because the initial linear momentum is zero, the two electrons must move in opposite directions with the value of the momentum. We assume everything is in one dimension for simplicity, and the momentum state for each electron is l k)l and l +k)2, respectively. We change the language now, for reasons that will become obvious below, such that Ik, )l > Ik, )A and Ik, )2 > lk, )3 Where A and B stand for receivers Alice and Bob. The state vector for this system is then given by |>= i2 (l k. milk. 12>}; | mama an) (13-11) J The subscripts are redundant with the sign of k, but remind us that that particle is heading to Alice or Bob since it contains information. Think ofthe zeprojection ofthe spin corresponding to a 0 or 1, for example, corresponding to a quantum bit (qubit) of information. Alice and Bob each have a measurement apparatus that determines if the particle they detect is spin tip or spin down. IfAlice measures T2 then using the above measurement prescription, the state vector becomes 1 x5, Wm,\" | imamk. m k. wk. m) dammit-.12).; (13.12) l-k.T2>A B - 1 - K, +=>Alk, tz)B) 1 8 + ) = = (1 - K, 1 = ) Alk, 1 = ) B + 1 - K, += >Alk, 4=>B)13.3 QUANTUM ENTANGLEMENT AND THE IMPACT OF MEASUREMENT 1 1 x ) = = (142)+ 112>) (13.15) V2 14 x ) = - 1 ( 14 2 ) - 112)) (13.16) with corresponding eigenvalues + - h. The question now is what does the state vector in Eq. 13.11 look like written in the x-basis? To see that, we have to now expand the z-basis states in the x-basis states, using Eqs 13.15 and 13.16 and find 14 2 ) = - ( 11 x ) + 14x) ) (13.17) 1 1 2 ) = - (13.18) VZ (It x ) - 14x)) Problem 13.1 Find the projection of [12) on |4x). What is the probability of finding this in the lab. So, we insert this into Eq. 13.11: 12) = - 1 - V2 ( 1 - K, 1 2 ) Al K , 4 2 ) B - 1 - K, 42 ) Alk, 12 )B) = 1 = 2V/2 ( (1 - k, 1x ) A - 1 - k, tx)A) ( /k, 1x)B + (K, hx)B) - (1 - k, 1 x ) A + 1 - k, tx)A) ( 1k, 1x) B - 1K, tx)B)) = ( 1 - k, 1 x ) A l k, 1 x ) B + 1 - K, 1 x ) Al k, h x ) B - 1 - K, t x ) Alk, 1 x ) B - 1 - K, tx)Alk, tx)B) 2V/2 ( 1 - k, 1 x ) Alk , 1 , ) B - 1 - k, Tax ) Alk, tx )B + 1 - k, tx)Alk, 1x)B - 1 - k, tx)Alk, tx) B) 2V/2 VZ (1 - k, 1x )Alk, 4x) B - 1 - K, Ix)Alk, 1x/B) (13.19) This is a remarkable/incredible/astounding result. The correlation that was built into the z-basis remains when the measurement is flipped to the x-axis! This is what Einstein and many others thought had to be wrong. It is remarkable that particle B follows the correlation set by the measurement of A, even though the decision to make the measurement in the x-bases was done after the particles were correlated in the z-basis and on their way. This is called the Einstein, Podolsky, and Rosen (EPR) paradox. This quantum property is incorporated in one or more quantum encryption protocols for secure transfer of information. A key step is demonstrating quantum entangled states, a critical feature for quantum computing. Quantum entangled states enable a quantum computer to outperform a classical computer for Shore's factoring algorithm, because the size of the quantum computer scales more slowly with the size of the number to be factored than the size of a classical computer. The size of a classical13 QUANTUM MEASUREMENT AND ENTANGLEMENT: WAVE FUNCTION COLLAPSE computer scales exponentially with the size of the number. Proof that states are entangled requires us to demonstrate that the correlations are preserved in two separate bases and satisfy Bell's inequality, which is discussed elsewhere.' Problem 13.2 Show that if Alice makes a measurement in the x-basis that the information originally then sent to Bob is lost in the z-basis. In other words, show that while Alice would have known with certainty the answer Bob would get, if she measures in the x-basis and forgets to tell Bob to rotate, she no longer knows what answer Bob will get and that Bob can measure either spin up or spin down